Abstract
The curved βγ system is a nonlinear σmodel with a Riemann surface as the source and a complex manifold X as the target. Its classical solutions pick
out the holomorphic maps from the Riemann surface into X. Physical arguments
identify its algebra of operators with a vertex algebra known as the chiral differential operators (CDO) of X. We verify these claims mathematically by constructing and quantizing rigorously this system using machinery developed by Kevin Costello and the second author, which combine renormalization, the BatalinVilkovisky formalism, and factorization algebras. Furthermore, we find that the factorization algebra of quantum observables of the curved βγ system encodes the sheaf of chiral differential operators. In this sense our approach provides deformation quantization for vertex algebras. As in many approaches to deformation quantization, a key role is played by GelfandKazhdan formal geometry. We begin by constructing a quantization of the βγ system with an ndimensional formal disk as the target. There is an obstruction to quantizing equivariantly with respect to the action of formal vector fields Wn on the target disk, and it is naturally identified with the first Pontryagin class in GelfandFuks cohomology. Any trivialization of the obstruction cocycle thus yields an equivariant quantization with respect to an extension of Wn by Ωb2cl, the closed 2forms on the disk. By machinery mentioned above, we then naturally obtain a factorization algebra of quantum observables, which has an associated vertex algebra easily identified with the formal βγ vertex algebra. Next, we introduce a version of GelfandKazhdan formal geometry suitable for factorization algebras, and we verify that for a complex manifold X with trivialized first Pontryagin class, the associated factorization algebra recovers the vertex algebra of CDOs of X.
out the holomorphic maps from the Riemann surface into X. Physical arguments
identify its algebra of operators with a vertex algebra known as the chiral differential operators (CDO) of X. We verify these claims mathematically by constructing and quantizing rigorously this system using machinery developed by Kevin Costello and the second author, which combine renormalization, the BatalinVilkovisky formalism, and factorization algebras. Furthermore, we find that the factorization algebra of quantum observables of the curved βγ system encodes the sheaf of chiral differential operators. In this sense our approach provides deformation quantization for vertex algebras. As in many approaches to deformation quantization, a key role is played by GelfandKazhdan formal geometry. We begin by constructing a quantization of the βγ system with an ndimensional formal disk as the target. There is an obstruction to quantizing equivariantly with respect to the action of formal vector fields Wn on the target disk, and it is naturally identified with the first Pontryagin class in GelfandFuks cohomology. Any trivialization of the obstruction cocycle thus yields an equivariant quantization with respect to an extension of Wn by Ωb2cl, the closed 2forms on the disk. By machinery mentioned above, we then naturally obtain a factorization algebra of quantum observables, which has an associated vertex algebra easily identified with the formal βγ vertex algebra. Next, we introduce a version of GelfandKazhdan formal geometry suitable for factorization algebras, and we verify that for a complex manifold X with trivialized first Pontryagin class, the associated factorization algebra recovers the vertex algebra of CDOs of X.
Original language  English 

Pages (fromto)  viii+210 
Number of pages  211 
Journal  Astérisque 
Volume  419 
DOIs  
Publication status  Published  14 Jun 2020 
Bibliographical note
ISBN 9782856299197This work would not have been possible without the support of several organizations. First, it was the open and stimulating atmosphere of the Max Planck Institute for Mathematics that made it so easy to begin our collaboration. Moreover, it is through the MPIM’s great generosity that we were able to continue work and finish the paper during several visits by VG and BW. Second, we benefited from the support and convivial setting of the Hausdorff Institute for Mathematics and its Trimester Program “Homotopy theory, manifolds, and field theories” during the summer of 2015. Third, the Oberwolfach Workshop “Factorization Algebras and Functorial Field Theories” in May 2016 allowed us all to gather in person and finish important discussions. In addition, OG enjoyed support from the National Science Foundation as a postdoctoral fellow under Award DMS1204826, and BW enjoyed support as a graduate student research fellow under Award DGE1324585. Finally, this research was carried out, in part, within the HSE University Basic Research Program and funded by the Russian Academic Excellence Project 5–100.
For OG there is a large cast of mathematicians whose questions, conversation,
and interest have kept these issues alive and provided myriad useful insights that are now hard to enumerate in detail. He thanks Kevin Costello for introducing him to the βγ system in graduate school—and for innumerable discussions since—as well as Dan BerwickEvans, Ryan Grady, and Yuan Shen for grappling collaboratively with [15] throughout that period. Si Li’s many insights and questions have shaped this work substantially. Matt Szczesny’s guidance at the Northwestern CDO Workshop was crucial; his subsequent encouragement is much appreciated. OG would also like to thank Stephan Stolz and Peter Teichner for the stillrunning conversation about conformal field theory that influences strongly his approach to the subject. Finally, he thanks André Henriques, John Francis, and Scott Carnahan for letting him eavesdrop as they chatted about CDOs over a decade ago.
BW feels fortunate to have stepped into this community early in his graduate
work and has benefited from the support of many of the individuals mentioned above. First and foremost, he thanks his adviser Kevin Costello for guidance and Si Li for helping him to harness Feynman diagrams in the context of the BV formalism. He also thanks Ryan Grady, Matt Szczesny, and Stephan Stolz for invitations to talk about this project as well as valuable input on various aspects of it. In addition, numerous discussions with Dylan William Butson, Chris Elliott, and Philsang Yoo about perturbative QFT have informed his work. Finally, we would like to thank Matt Szczesny and James Ladouce for pointing out numerous typos and providing feedback on an earlier draft of this paper.
Keywords
 GanGrossPrasad conjecture
 Local trace formula
 Padic Lie groups
 Representations of real
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Vasily Gorbunov
 School of Natural & Computing Sciences, Mathematical Science  Emeritus Professor
Person: Honorary